From VOAs to Conformal Nets

A simple strongly local unitary VOA $V$ canonically generates an irreducible conformal net $\mathcal A_V$ on the Hilbert completion of $V$: the local algebra of an interval is generated by closed smeared vertex operators supported there. The theorem retains the vacuum Hilbert space, positive-energy conformal symmetry, interval localization, and—within this strongly local construction—the VOA can be recovered from the net. It does not apply to a unitary VOA for which strong locality has not been proved.

Required background. Unitary VOAs, Energy Bounds, and Strong Locality supplies the Hilbert completion. Conformal Nets and Covariance Axioms supplies the target axioms. Energy Bounds, Strong Locality, and Smeared Vertex Operators supplies domain and closure control.

Helpful background. DHR Sectors and Modular Tensor Categories of Nets explains why comparing module and sector categories is an additional theorem.

The construction theorem

Let $V$ be simple, unitary, energy-bounded, and strongly local. Let $\mathcal H_V$ be its Hilbert completion and $\Omega=\mathbf1$. For every nonempty nondense interval $I\subset S^1$, set

$$\mathcal A_V(I)=W^*\{Y(v,f):v\in V, f\in C^\infty(S^1),\ \operatorname{supp}f\subset I\}.$$

Then $I\mapsto\mathcal A_V(I)$ is an irreducible diffeomorphism-covariant conformal net Carpi et al. 2018, Theorem 6.8, PDF pp. 52–53. Each hypothesis has a distinct role:

• unitarity gives a positive Hilbert completion and adjoints;
• polynomial energy bounds define closable smeared fields on a common invariant core;
• strong locality gives $\mathcal A_V(I)\subset\mathcal A_V(I')'$;
• simplicity and $V_0=\mathbb C\Omega$ give the irreducible vacuum representation;
• the Virasoro field integrates to positive-energy diffeomorphism covariance.

Isotony follows immediately by enlarging support. The vacuum is invariant under conformal transformations and cyclic for the global algebra by the creation property and density of finite-energy states. Positive energy is inherited from $L_0\geq0$. Locality is not re-proved from the Jacobi identity at this stage; it is exactly the strong-locality input.

The construction is independent, up to net isomorphism, of the normalized invariant scalar product chosen on a simple strongly local VOA, as included in the same theorem. Unit-preserving automorphisms compatible with the scalar product act on the net, but identifying complete automorphism groups or representation categories requires the corresponding later results.

The U(1) current net

The first application returns to Free Bosons and Vertex Operators. For the rank-one Heisenberg VOA, the Hilbert completion is the bosonic Fock space and real smeared currents $J(f)$ are essentially self-adjoint. One may equivalently generate

$$\mathcal A_{U(1)}(I) =\{e^{i\overline{J(f)}}:\operatorname{supp}f\subset I,\ f\text{ real}\}''.$$

If $I_1\subset I_2$, the set of allowed test functions is nested, proving isotony. For disjoint intervals, the current symplectic form vanishes, so the Weyl operators commute and give locality. Circle diffeomorphisms transport supports and the vacuum representation has nonnegative rotation generator. The Fock vacuum is cyclic because derivatives at the identity of products of Weyl operators create the oscillator states, which are dense. These checks identify the generated object as the U(1) current conformal net rather than merely an abstract family of algebras.

An independent recovery check uses the finite-energy subspace. The vector $a_{-1}\Omega$ is recovered as the weight-one current state; its Fredenhagen–Jörß field agrees with $J(f)$. Normal-ordered products of the recovered current rebuild the Heisenberg state–field correspondence. In general, Carpi and collaborators prove this round trip for a strongly local simple unitary VOA: the Fredenhagen–Jörß smeared fields of $\mathcal A_V$ coincide with the original smeared vertex operators Carpi et al. 2018, Theorem 9.2, PDF pp. 67–68.

The recovery method itself starts from modularly defined fields affiliated with local algebras and controlled by conformal energy Fredenhagen and Jörß 1996, §§2–3, pp. 545–550. The equality theorem is therefore substantive: it identifies two independently defined operator-valued distributions on a common finite-energy core, rather than simply reading the original formal series back from notation.

What is and is not retained

The construction retains $\mathcal H_V$, $\Omega$, $L_0$, the projective conformal action, and interval von Neumann algebras. Within the theorem’s image, it retains enough localized finite-energy fields to recover $V$. It does not automatically identify every algebraic $V$-module with a DHR representation of $\mathcal A_V$, prove complete rationality, or show that every extension or coset is preserved. Those comparisons demand integrability, finite-index, or categorical hypotheses.

Adversarial input

Take a unitary VOA with formal locality but no established polynomial energy bounds or strong locality. Writing the same formula for $\mathcal A_V(I)$ does not license a conformal net: the smeared series may lack a controlled closure, and disjoint-support core commutators may fail to imply commuting spectral projections. A nonunitary logarithmic VOA fails even earlier because the positive Hilbert-space target and adjoint structure are absent. In both cases the formal VOA remains valid; the net theorem is simply inapplicable.

Exercises

1. Prove covariance implies isotony is preserved under interval transport.

Solution

If $I_1\subset I_2$, then $\gamma I_1\subset\gamma I_2$. Covariance gives $U(\gamma)\mathcal A(I_j)U(\gamma)^*=\mathcal A(\gamma I_j)$, so conjugating the original inclusion yields the transported one.

2. Why is equality of current commutators on finite-particle states insufficient without the Weyl argument?

Solution

The local algebras depend on closed operators and their spectral projections. Core commutation need not imply strong commutation. The Weyl relations are identities of bounded unitaries, so they directly prove that the generated von Neumann algebras commute.

References

• Carpi, Sebastiano, Yasuyuki Kawahigashi, Roberto Longo, and Mihály Weiner. “From Vertex Operator Algebras to Conformal Nets and Back.” Communications in Mathematical Physics 364 (2018), 101–145. DOI. Open PDF.
• Fredenhagen, Klaus, and Martin Jörß. “Conformal Haag–Kastler Nets, Pointlike Localized Fields and the Existence of Operator Product Expansions.” Communications in Mathematical Physics 176 (1996), 541–554. DOI.